Vaccinated people carry similar viral loads to unvaccinated people and can transmit the virus to others.

FDA Comirnaty Q & A:

“the scientific community does not yet know if the Pfizer-BioNTech COVID-19 Vaccine or Comirnaty will reduce such transmission.”
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FDA Comirnaty Q & A:

“Based on a small number of severe cases in the data set submitted for the EUA, it was difficult to evaluate whether the Pfizer-BioNTech COVID-19 Vaccine reduces the severity of COVID-19. ”
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FDA Comirnaty Q & A: Q:

Does the vaccine protect against asymptomatic SARS-CoV-2 infection? (Asymptomatic infection is when someone is infected with SARS-CoV-2 but does not have signs or symptoms of COVID-19.) A: It is not known if the vaccine protects against asymptomatic SARS-CoV-2 infection.
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“We know the efficacy of vaccination drops and that infection probability is very similar to that of the unvaccinated population after five or six months”
Prof. Eli Waxman of the Weizmann Institute of Science, Was Israel’s decision to give everyone COVID boosters a shot in the dark?

Study: No Significant Difference in Viral Load Between Vaccinated and Unvaccinated, Asymptomatic and Symptomatic Groups Infected with SARS-CoV-2 Delta Variant

Article: Study: Fully Vaccinated Healthcare Workers Carry 251 Times Viral Load, Pose Threat to Unvaccinated Patients, Co-Workers
Can You Spread COVID After Receiving Vaccine? Short answer: YES
This study was published in July 2021: Vaccinated and unvaccinated individuals have similar viral loads in communities with a high prevalence of the SARS-CoV-2 delta variant